Polynomial 2D Green Coordinates for Polygonal Cages

ACM Transaction on Graphics (SIGGRAPH ' Conference Proceedings)
√Člie Michel
Jean-Marc Thiery
Teaser image
Top row: (a) input image and an embedding polygonal cage; (b-d) deformations obtained using Mean-Value coordinates, Cubic Mean-Value coordinates, and Green coordinates; (e) our conformal deformations obtained with cubic curves. Bottom row: more resuls of our approach, using polynomial curves of various orders (from 1 to 7).

Abstract

Cage coordinates are a powerful means to define 2D deformation fields from sparse control points. We introduce Conformal polynomial Coordinates for closed polyhedral cages, enabling segments to be transformed into polynomial curves of any order. Extending classical 2D Green coordinates, our coordinates result in conformal harmonic deformations that are cage-aware. We demonstrate the usefulness of our technique on a variety of 2D deformation scenarios where curves allow artists to perform intuitive deformations with few input parameters. Our method combines the texture preservation property of conformal deformations together with the expressiveness offered by Bezier controls.

Video

Examples

Left: Straight Green Coordinates. Right: Cubic Green Coordinates (ours).

Left: Straight Green Coordinates. Right: Cubic Green Coordinates (ours).

Left: Straight Green Coordinates. Right: Cubic Green Coordinates (ours).

Citation

@article{michel23polynomial, title = {Polynomial 2D Green Coordinates for Polygonal Cages}, author = {Michel, √Člie and Thiery, Jean-Marc}, journal = {ACM Transaction on Graphics (SIGGRAPH '23 Conference Proceedings)}, year = {2023}, publisher = {ACM}, doi = {10.1145/3588432.3591499}, }